The diminutive forward relies on his speed and skating ability and was praised by Hitmen Head Coach Mike Williamson for his competitive nature and willing to play any role the team asks of him.[8] Williamson added that his sense of anticipation allows Kozun to see plays on the ice develop before they happen, stating it to be a trait found in elite athletes.[2] His small size, standing 5 feet 9 inches (1.75 m), initially led WHL teams to overlook him,[2] as well as NHL scouts who argued he was too small for the NHL.[9] He was left unranked by NHL Central Scouting in his first year of draft eligibility in 2008,[10] and went undrafted that year. Following his 108-point season with the Hitmen, the Los Angeles Kings selected Kozun with their sixth-round pick, 179th overall, at the 2009 NHL Entry Draft.[8] Kozun led the WHL in scoring in 2009–10, scoring 13 points in his final five games to finish with 107 points to win the Bob Clarke Trophy.[11]

In his home debut for the Monarchs, on October 16, 2010, against the Providence Bruins, Kozun scored two goals, the first of which was a penalty shot goal at 19:26 of the game's first period. The goal was the first of two penalty shot attempts by the Monarchs, a franchise record for penalty shots awarded in a game.

After four seasons in the Kings organization, Kozun was traded to the Toronto Maple Leafs on January 22, 2014, in exchange for forward Andrew Crescenzi. Kozun was assigned to the AHL's Toronto Marlies.[13] After an impressive display in pre-season Kozun made the final cut for the Maple Leafs for the 2014–15 season. He scored his first NHL point with an assist on a goal by Nazem Kadri in the season opener against the Montreal Canadiens on October 8, 2014. He was later placed on waivers on December 17, 2014.[14] He would remain with the organization, however, and later in the season, on February 20, 2015, Kozun scored his first career NHL goal in his eighth game played.

After three seasons with Lokomotiv Yaroslavl, Kozun left as a free agent following the 2018–19 season. He signed a one-year contract to continue in the KHL with Metallurg Magnitogorsk on May 1, 2019.[17]

Kozun earned the attention of both the Canadian and American national junior teams whilst leading the WHL in scoring early in the 2009–10 season.[18] Kozun was reportedly guaranteed a spot on the American team at the 2010 World Junior Ice Hockey Championships,[4] but declined as he wanted to show loyalty to the nation that developed him as a hockey player.[19] He later earned a spot on the Canadian team, his first time representing Hockey Canada internationally.[8] During round robin play, he scored the shootout winner to cap a dramatic 5–4 come from behind victory over the Americans, leading Canada to first place in their pool, and a bye into the semi-final round of the tournament.[20] Canada, however, lost a rematch against the Americans in the gold medal game, also in overtime, to settle for the silver medal.